multiscale modeling of brittle composites using reduced

Some reduced order models based on transformation field analysis (e.g. lead to spurious residual stress fields at microscale failure. Section 3 gives a reduced order symmetric model formulation for the microscale problem).

Microscale Problem

The drag and displacement jump components are expressed in terms of a local coordinate system formed by the normal and tangential directions at the interface. The present formulation is limited to cases characterized by a small magnitude of the RVE compared to the length of the strain and stress waves.

Macroscale Problem

The governing equations and discrete approximations of the elastic impact and damage functions are given in Ref. 23, pre-multiplying the resulting equation by Nint(α)(ˆy) and integrating over the domain of RVE gives (α.

Reduced-Order Model Development Strategy

The rest of this section discusses a new strategy for selecting the model order and partitioning the RVE domain, as well as the numerical evaluation of the reduced-order model. The model selection strategy consists of identifying the failure paths in the microstructure when subjected to a series of loading conditions and partitioning the domain of the RVE as well as the interfaces by selecting each failure path as a partition.

Figure 2: The partitioning and model reduction strategy. Failure profiles within the RVE subjected to (a) uniform biaxial loading; (b) uniaxial in the lateral direction;

Numerical Evaluation of the Reduced-Order Model

Return to the start of the iteration loop (e) Update the eigendeformation vector: d=kd (f) Update the active set: A=kW. g) Exit the algorithm End iteration loop. Box 1: The active set algorithm for evaluating the reduced order model with one-sided contact constraints.

Two-Order Reduced Modeling

Map the damage coefficients from the low-order model to the high-order model partitions. In this study, each finite element within the RVE domain forms a partition for the higher order model.

RVE Analysis

The values ​​of the model parameters used in the reduced order model differ from those of the direct numerical simulations. The purpose of the proposed reduced order model is to capture the failure mechanisms within the heterogeneous material in a computationally efficient and accurate manner.

Figure 3: Stress-strain response and damage evolution within the RVE when sub- sub-jected to uniform biaxial loading.

Crack Propagation in a Beam Subjected to Three-Point Bending

Figures 7a and 7b illustrate the propagation of the initial matrix crack and damage in the matrix as predicted by the direct numerical simulation and the SBU-4-6 model. A comparison of the reaction force applied displacement curves for the numerical simulation and the proposed multiscale model is shown in Fig. 10a and 10b show the failure of the three-point bending plate in the presence of interface effects.

The comparison of the applied force-deflection curve predicted with the proposed multiscale model and the direct numerical simulations is shown in Fig.

Figure 7: Damage profile of the 3-point bending beam specimen at the onset of shear fracture in the absence of interface debonding effects: (a) The prediction of the SBU-4-6 model

Multiple Scale Model

Macrochronological - macroscopic problem: Given: average body force, ¯b, limit data ˆu0 and ˆt0 and solution of macrochronological-microscopic. Macrochronological and microchronological problems are related to a quasi-periodic rate operator that defines the evolution of temporally homogenized response fields (i.e., equation 91b). The governing equations and discrete approximations of the influence functions induced by elastic and phase damage are given in ref.

The macrochronological counterparts of the reduced-order microscopic balance and the macroscopic stress are obtained by applying the quasi-periodic time homogenization operator to Eqs.

Cyclic Damage Model

In this section, we focus on the implementation details of the coupling between the micro- and macrochronological problems. The overall solution strategy for the evaluation of the coupled multiscale system is illustrated in Fig. A driver program (implemented in the Python programming language for compatibility with Abaqus) controls the execution of the solution procedure.

The macrochronological response fields provide the initial state of the microchronological - multiscopic problem at a fixed time ten due to the particular choice of the temporal homogenization operator.

Adaptive Macrochronological Time Stepping

This is due to the deviation of the damage accumulation from the piecewise quadratic approximation within the time step. The time step reduction is repeated until the desired accuracy of time to failure is achieved. The failure event is defined as a loss of load-bearing capacity of the structure, detected as lack of convergence occurring during the evaluation of the microchronological problem.

Structural failure is assumed to occur when the failure event is detected, and the ratio of the current macrochronological time step size and the current macrochronological time is less than a specified tolerance (toll) value.

Improved Adaptive Stepping Criterion

The proposed adaptive macrochronological time-stepping strategy is verified by comparing the performance with direct cycle-by-cycle simulations. The effectiveness of the proposed approach is further demonstrated by performing simulations when the unit cell is exposed to a slightly smaller load amplitude. In this analysis, the cycle to failure of the first partition is 677 (as opposed to 377 in the previous simulations).

The total resolved cycles of the proposed adaptive time-stepping strategy remain largely the same (∆Dmax and 2% are 118, 72, and 47, respectively), indicating a significant computational advantage under high cycle error conditions.

Figure 14: Comparison of the cyclic damage accumulation computed using the direct cycle-by-cycle approach and proposed multiscale model with adaptive time stepping.

Improved Adaptive Stepping Criterion Verification

The capabilities of the proposed multiple spatio-temporal methodology are evaluated through the investigation of graphite fiber-reinforced epoxy composites (ie, IM7/977-3) subjected to cyclic loading. This section presents the experiments performed to study the cyclic response of the composite; calibration of model parameters based on monotonic and cyclic experiments, and; validation of model predictions based on acoustic emission testing.

Experiments

All non-zero layers in the quasi-isotropic samples exhibited matrix-dominated failure, while the zero layers exhibited fiber failure. A Poisson's ratio of 0.316 with a standard deviation of 0.039 was observed by placing a strain gauge perpendicular to the load on the 0◦ samples. The maximum applied stress amplitude for the 90◦ samples ranged between 45% and 55% of the average monotonic final stress of the layup configuration.

The quasi-isotropic layup was tested with a maximum applied stress amplitude of 17% of the ultimate stress of the corresponding layup.

Table 1: IM7/977-3 specimen dimensions Fiber Number Length Width Thickness Orientation Of Plies [mm] [mm] [mm]

Model Calibration

The values ​​of αmandαf are chosen to avoid numerical problems associated with sudden failure events while accurately capturing the characteristics of the stress-strain response. The experimentally observed stress-strain response of the zero and ninety degree specimens is used in the calibration process. The calibration of the cyclic load sensitivity parameters is performed using the stress-life curves obtained from experiments in which ninety degree specimens are subjected to cyclic load in the tensile direction.

In the experiments, the maximum cyclic load amplitude varied between 360 MPa.

Table 3: Elastic parameter optimization

Model Validation

The orientation of the layer in which the failure event occurs is also shown in Figure 22. All the failure events were in the matrix with the exception of the final event. We further assessed the validity of the proposed model by comparing the model predictions with experiments on quasi-isotropic specimens subjected to cyclic loading conditions.

Despite the close correlation with acoustic emission testing, the predictive ability of the proposed model is qualitative.

Figure 21: Comparison of experimental and predicted stress-strain curves of the quasi- quasi-isotropic specimens subjected to monotonic tensile loading.

Material Fabrication

Testing

Acoustic Emission

X-ray Radiography

X-ray Computed Tomography

Computational Model

Each layer was explicitly modeled along the thickness direction with 16 elements discretizing the thickness of the sample layup. Only the top half of the sample was discretized due to the symmetry of the sample. A small part of the specimen along the length (L=6 mm) was modeled to reduce the computational cost of the failure simulation.

The sample was loaded by increasing the average distance between the top and bottom edges using constraints.

Multiscale Failure Modeling

In this study, we use the eigendeformation-based reduced-order homogenization method with symmetric coefficients (EHM) [17] to effectively evaluate the response at the scale of the microstructure. The unit cell of the CFRP composite material in a single layer is shown in Fig. Consider the division of the unit cell domain into n parts, within which the strains and damage are assumed to be spatially constant.

The breakdown of the unit cell used in the present study is shown in Fig.

Figure 25: The unit cell for IM7/977-3 with 66% fiber volume fraction.

Calibration of the Model Parameters

The constituent elastic moduli, E(m), E1(f), E2(f) and G(f)12, were calibrated by minimizing the discrepancy between the composite elastic moduli of 0◦ and 90◦ samples and the simulated elastic moduli of the homogenized composite. The damage accumulation parameters are calibrated based on sets of experiments performed on unidirectional 0◦ and 90◦ specimens. The constitutive parameters are identified by minimizing the discrepancy between experimentally observed stress–strain response and numerical predictions of the multiscale model in a least-squares sense.

The calibrated and experimentally observed stress-strain response of the 0◦ and 90◦ samples is shown in Fig.

Results and Discussion

Extensive 45◦ and 90◦ cracks and delaminations along the length of the specimen are evident. Matrix cracks initiated at ±45◦ inserts propagate rapidly along the length of the specimen. Die cracking in 0◦ inserts remains negligible until the load reaches close to the ultimate failure strength of the specimen.

The edge delaminations continue to grow slowly until the strain reaches close to the ultimate strength of the specimen.

Figure 28: The stress-strain response of the calibrated model compared with experi- experi-mental data: (a) specimens with 90 ◦ layup and (b) specimens with 0 ◦ layup.

Computational Model

Multiscale Failure Modeling

D(α) ≤1 (133) where p is the cyclic susceptibility parameter, h·i+ denotes the MacCauley brackets, Φ is the damage evolution law for monotonic loading, and υ(α) is the damage equivalent strain defined as. 2ε(α) :L(α) :ε(α) (134) The arctangent law, which is used for monotonically loaded samples, is considered to be the monotonic law of damage development. The division of the unit cell used in this investigation is the same as that used for monotonic loads.

Calibration of Model Parameters

Results and Discussion

40 shows transverse matrix cracks in the seventh layer from the top of the sample, which also has an orientation of -45◦. At 1500 loading cycles, there were no cracks in the sample, which reflected the behavior of the experimental samples. The simplest method of applying periodic boundary conditions is to obtain a periodic mesh of the RVE.

Next, the combined mesh was copied and reflected three times to create a periodic surface mesh of the matrix region of the RVE.

Figure 37: The calibrated fatigue response of unidirectional 90 ◦ specimens is com- com-pared with a statistical curve fit of the experimental determined fatigue response.